C₂₃×#10624595 #30661102

order	16767
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₂₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₂₃×#10624595

Subgroups

	subgroup	G/H
center, socle	C₃²×C₂₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₂₃
prime radical	C₂₃×#6856	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₂₃ ◁ C₃²×C₉×C₂₃ ◁ G

318 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 5589

A minimal faithful permutation representation — 3 generators on 59 points:

(2,4,6)(10,11,14)(12,18,28)(13,15,23)(16,24,32)(17,27,21)(19,30,34)(20,26,22)(25,35,31)(29,33,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59);(1,2,5)(3,6,9)(4,8,7)(11,15,17)(16,18,26)(25,29,30);(10,12,19,13,20,31,21,32,36)(11,16,25,15,18,29,17,26,30)(14,22,33,23,24,34,27,28,35)

Conjugacy classes & character table

1679 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16767, 155);